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Theorem opelopabsb 4275
Description: The law of concretion in terms of substitutions. (Contributed by NM, 30-Sep-2002.) (Revised by Mario Carneiro, 18-Nov-2016.)
Assertion
Ref Expression
opelopabsb  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [. A  /  x ]. [. B  / 
y ]. ph )
Distinct variable groups:    x, y    x, B
Allowed substitution hints:    ph( x, y)    A( x, y)    B( y)

Proof of Theorem opelopabsb
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2791 . . . . . . . . . 10  |-  x  e. 
_V
2 vex 2791 . . . . . . . . . 10  |-  y  e. 
_V
31, 2opnzi 4243 . . . . . . . . 9  |-  <. x ,  y >.  =/=  (/)
4 simpl 443 . . . . . . . . . . 11  |-  ( (
(/)  =  <. x ,  y >.  /\  ph )  ->  (/)  =  <. x ,  y >. )
54eqcomd 2288 . . . . . . . . . 10  |-  ( (
(/)  =  <. x ,  y >.  /\  ph )  ->  <. x ,  y
>.  =  (/) )
65necon3ai 2486 . . . . . . . . 9  |-  ( <.
x ,  y >.  =/=  (/)  ->  -.  ( (/)  =  <. x ,  y
>.  /\  ph ) )
73, 6ax-mp 8 . . . . . . . 8  |-  -.  ( (/)  =  <. x ,  y
>.  /\  ph )
87nex 1542 . . . . . . 7  |-  -.  E. y ( (/)  =  <. x ,  y >.  /\  ph )
98nex 1542 . . . . . 6  |-  -.  E. x E. y ( (/)  =  <. x ,  y
>.  /\  ph )
10 elopab 4272 . . . . . 6  |-  ( (/)  e.  { <. x ,  y
>.  |  ph }  <->  E. x E. y ( (/)  =  <. x ,  y >.  /\  ph ) )
119, 10mtbir 290 . . . . 5  |-  -.  (/)  e.  { <. x ,  y >.  |  ph }
12 eleq1 2343 . . . . 5  |-  ( <. A ,  B >.  =  (/)  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph } 
<->  (/)  e.  { <. x ,  y >.  |  ph } ) )
1311, 12mtbiri 294 . . . 4  |-  ( <. A ,  B >.  =  (/)  ->  -.  <. A ,  B >.  e.  { <. x ,  y >.  |  ph } )
1413necon2ai 2491 . . 3  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  ->  <. A ,  B >.  =/=  (/) )
15 opnz 4242 . . 3  |-  ( <. A ,  B >.  =/=  (/) 
<->  ( A  e.  _V  /\  B  e.  _V )
)
1614, 15sylib 188 . 2  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  ->  ( A  e.  _V  /\  B  e.  _V )
)
17 sbcex 3000 . . 3  |-  ( [. A  /  x ]. [. B  /  y ]. ph  ->  A  e.  _V )
18 spesbc 3072 . . . 4  |-  ( [. A  /  x ]. [. B  /  y ]. ph  ->  E. x [. B  / 
y ]. ph )
19 sbcex 3000 . . . . 5  |-  ( [. B  /  y ]. ph  ->  B  e.  _V )
2019exlimiv 1666 . . . 4  |-  ( E. x [. B  / 
y ]. ph  ->  B  e.  _V )
2118, 20syl 15 . . 3  |-  ( [. A  /  x ]. [. B  /  y ]. ph  ->  B  e.  _V )
2217, 21jca 518 . 2  |-  ( [. A  /  x ]. [. B  /  y ]. ph  ->  ( A  e.  _V  /\  B  e.  _V )
)
23 opeq1 3796 . . . . 5  |-  ( z  =  A  ->  <. z ,  w >.  =  <. A ,  w >. )
2423eleq1d 2349 . . . 4  |-  ( z  =  A  ->  ( <. z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  <. A ,  w >.  e.  { <. x ,  y >.  |  ph } ) )
25 dfsbcq2 2994 . . . 4  |-  ( z  =  A  ->  ( [ z  /  x ] [ w  /  y ] ph  <->  [. A  /  x ]. [ w  /  y ] ph ) )
2624, 25bibi12d 312 . . 3  |-  ( z  =  A  ->  (
( <. z ,  w >.  e.  { <. x ,  y >.  |  ph } 
<->  [ z  /  x ] [ w  /  y ] ph )  <->  ( <. A ,  w >.  e.  { <. x ,  y >.  |  ph }  <->  [. A  /  x ]. [ w  / 
y ] ph )
) )
27 opeq2 3797 . . . . 5  |-  ( w  =  B  ->  <. A ,  w >.  =  <. A ,  B >. )
2827eleq1d 2349 . . . 4  |-  ( w  =  B  ->  ( <. A ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ph } ) )
29 dfsbcq2 2994 . . . . 5  |-  ( w  =  B  ->  ( [ w  /  y ] ph  <->  [. B  /  y ]. ph ) )
3029sbcbidv 3045 . . . 4  |-  ( w  =  B  ->  ( [. A  /  x ]. [ w  /  y ] ph  <->  [. A  /  x ]. [. B  /  y ]. ph ) )
3128, 30bibi12d 312 . . 3  |-  ( w  =  B  ->  (
( <. A ,  w >.  e.  { <. x ,  y >.  |  ph } 
<-> 
[. A  /  x ]. [ w  /  y ] ph )  <->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph }  <->  [. A  /  x ]. [. B  / 
y ]. ph ) ) )
32 nfopab1 4085 . . . . . 6  |-  F/_ x { <. x ,  y
>.  |  ph }
3332nfel2 2431 . . . . 5  |-  F/ x <. z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }
34 nfs1v 2045 . . . . 5  |-  F/ x [ z  /  x ] [ w  /  y ] ph
3533, 34nfbi 1772 . . . 4  |-  F/ x
( <. z ,  w >.  e.  { <. x ,  y >.  |  ph } 
<->  [ z  /  x ] [ w  /  y ] ph )
36 opeq1 3796 . . . . . 6  |-  ( x  =  z  ->  <. x ,  w >.  =  <. z ,  w >. )
3736eleq1d 2349 . . . . 5  |-  ( x  =  z  ->  ( <. x ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  <. z ,  w >.  e.  { <. x ,  y >.  |  ph } ) )
38 sbequ12 1860 . . . . 5  |-  ( x  =  z  ->  ( [ w  /  y ] ph  <->  [ z  /  x ] [ w  /  y ] ph ) )
3937, 38bibi12d 312 . . . 4  |-  ( x  =  z  ->  (
( <. x ,  w >.  e.  { <. x ,  y >.  |  ph } 
<->  [ w  /  y ] ph )  <->  ( <. z ,  w >.  e.  { <. x ,  y >.  |  ph }  <->  [ z  /  x ] [ w  /  y ] ph ) ) )
40 nfopab2 4086 . . . . . . 7  |-  F/_ y { <. x ,  y
>.  |  ph }
4140nfel2 2431 . . . . . 6  |-  F/ y
<. x ,  w >.  e. 
{ <. x ,  y
>.  |  ph }
42 nfs1v 2045 . . . . . 6  |-  F/ y [ w  /  y ] ph
4341, 42nfbi 1772 . . . . 5  |-  F/ y ( <. x ,  w >.  e.  { <. x ,  y >.  |  ph } 
<->  [ w  /  y ] ph )
44 opeq2 3797 . . . . . . 7  |-  ( y  =  w  ->  <. x ,  y >.  =  <. x ,  w >. )
4544eleq1d 2349 . . . . . 6  |-  ( y  =  w  ->  ( <. x ,  y >.  e.  { <. x ,  y
>.  |  ph }  <->  <. x ,  w >.  e.  { <. x ,  y >.  |  ph } ) )
46 sbequ12 1860 . . . . . 6  |-  ( y  =  w  ->  ( ph 
<->  [ w  /  y ] ph ) )
4745, 46bibi12d 312 . . . . 5  |-  ( y  =  w  ->  (
( <. x ,  y
>.  e.  { <. x ,  y >.  |  ph } 
<-> 
ph )  <->  ( <. x ,  w >.  e.  { <. x ,  y >.  |  ph }  <->  [ w  /  y ] ph ) ) )
48 opabid 4271 . . . . 5  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ph }  <->  ph )
4943, 47, 48chvar 1926 . . . 4  |-  ( <.
x ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [ w  /  y ] ph )
5035, 39, 49chvar 1926 . . 3  |-  ( <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [ z  /  x ] [ w  /  y ] ph )
5126, 31, 50vtocl2g 2847 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph } 
<-> 
[. A  /  x ]. [. B  /  y ]. ph ) )
5216, 22, 51pm5.21nii 342 1  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [. A  /  x ]. [. B  / 
y ]. ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623   [wsb 1629    e. wcel 1684    =/= wne 2446   _Vcvv 2788   [.wsbc 2991   (/)c0 3455   <.cop 3643   {copab 4076
This theorem is referenced by:  brabsb  4276  opelopabaf  4288  opelopabf  4289  difopab  4817  isarep1  5331
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078
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